Video hosting is always a pressure on the CPU and RAM of the web hosting service you are using. In this matter I think you need to start your search from VPs solutions only.
I have some home videos hosted on the VPS which I have from pinellashosting.com.
$15 is not a fortune for the such kind of the hosting.
VPS is better in comparision with other so called media hosts (which are shared and have the same price )

As you are planning to launch video website so I guess you are going to use ffMPEG or PHPMotion script. To run all these applications, you need a well configured VPS or dedicated server. If you are going to use VPS, then make sure it is based on hardware virtualization.

If you plan on hosting the videos then be prepared for huge bandwidth increases, expecially if you get alot of visitors. A website with only a thousand visitors a day will use up your server resources and bandwith quickly. Be sure to optimize the videos to keep the file size as low as possible. Convert the video in MP4 is the format to keep the file size little and retain a quality view.
One way we found to be cost effective is hosting the videos at Amazon S3 and inlaying them on your site. The bandwidth costs are very inexpensive and the network is awesome at Amazon.
For high scale video hosting, Voxel out of New York is one I recommend to my corp clients.